A211183 Triangle T(n,k), 0<=k<=n, read by rows, given by (0, 1, 1, 3, 3, 6, 6, 10, 10, 15, ...) DELTA (1, 0, 2, 0, 3, 0, 4, 0, 5, ...) where DELTA is the operator defined in A084938.

1, 0, 1, 0, 1, 1, 0, 2, 4, 1, 0, 7, 19, 11, 1, 0, 38, 123, 107, 26, 1, 0, 295, 1076, 1195, 474, 57, 1, 0, 3098, 12350, 16198, 8668, 1836, 120, 1, 0, 42271, 180729, 268015, 176091, 52831, 6549, 247, 1, 0, 726734, 3290353, 5369639, 4105015, 1564817, 287473, 22145

Offset

0,8

Links

Paul D. Hanna, Rows n = 0..31, flattened.

Formula

Sum_{k, 0<=k<=n}T(n,k)*x^(n-k) = A000012(n), A000366(n+1), A110501(n+1), A211194(n), A221972(n) for x = 0, 1, 2, 3, 4 respectively.

T(n,n-1) = A000295(n).

T(n,1) = A000366(n).

G.f.: A(x,y) = Sum_{n>=0} n! * x^n * Product_{k=1..n} (y + (k-1)/2) / (1 + (k*y + k*(k-1)/2)*x). - Paul D. Hanna, Feb 03 2013

Example

Triangle begins :
1;
0, 1;
0, 1, 1;
0, 2, 4, 1;
0, 7, 19, 11, 1;
0, 38, 123, 107, 26, 1;
0, 295, 1076, 1195, 474, 57, 1;
0, 3098, 12350, 16198, 8668, 1836, 120, 1;
0, 42271, 180729, 268015, 176091, 52831, 6549, 247, 1;
0, 726734, 3290353, 5369639, 4105015, 1564817, 287473, 22145, 502, 1; ...

Prog

(PARI) T(n, k)=polcoeff(polcoeff(sum(m=0, n, m!*x^m*prod(k=1, m, (y + (k-1)/2)/(1+(k*y+k*(k-1)/2)*x+x*O(x^n)))), n, x), k, y)
for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print()) \\ Paul D. Hanna, Feb 03 2013

Crossrefs

Cf. A000366, A110501, A211194, A221972.

Sequence in context: A294390 A152433 A094344 * A137391 A276207 A248672

Adjacent sequences: A211180 A211181 A211182 * A211184 A211185 A211186

Keyword

nonn,tabl

Author

Philippe Deléham, Feb 02 2013

Status

approved